An NL Fragment for Inclusion Logic Dietmar Berwanger joint work - PowerPoint PPT Presentation

An NL Fragment for Inclusion Logic Dietmar Berwanger joint work with Erich Grädel Dagstuhl, June  Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 1 / 10

Non-reachability of Q NL ∃ z ( x ⊆ z ∧ ¬ Qz ∧ ∀ y ( Ezy → y ⊆ z )) Winning safety game within Q P-complete Qx ∧ ∃ z ( z ⊆ x ∧ ( V  → ∃ y ( Exy ∧ y ⊆ z )) ∧ ( V  → ∀ y ( Exy → y ⊆ z )) Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 2 / 10

Solitaire games and GFP Solitaire games — only one player has nontrivial moves, in every SCC Inspire restriction to Solitaire GFP : - negation: only fixed-point-free formulae - conjunction: one member fixed-point-free Captures NL on finite structures & more expressive than FO + TC on arbitrary structures Goal: Find the Solitaire-GFP fragment of Inclusion logic Obstacles: closed formulae in team semantics? dual of safety conditions Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 3 / 10

Flattening Definition Formula φ ∈ Inc is flat : A ⊧ X φ iff A ⊧ { s } φ , for all s ∈ X . Lemma: flattening φ ( ¯ x ) is flat iff for any structure A , team X A ⊧ X φ ( ¯ x ) ⇐ ⇒ A ⊧ X ∃ ¯ z ( ¯ x ⊆ ¯ z ∧ φ ( ¯ z )) �ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ�ÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜÜ� φ ♭ ( ¯ x ) Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 4 / 10

Te fragment. Take  F ∶∶= literals, ⊆ atoms φ ∨ η ∃ ¯ xφ for φ , η ∈ F φ ♭ ∧ η ∀ ¯ xφ ♭ ⎧ ⎪ ⎪ φ ( ¯ x ) if φ ∈ FO ⎪ φ ♭ ( ¯ ⎨ with x ) ∶= ⎪ ⎪ ∃ ¯ z ( ¯ x ⊆ ¯ z ∧ φ ( ¯ z )) otherwise ⎪ ⎩ Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 5 / 10

Te fragment. Take  – stratified S ∃  = S ∀  = S  ∶∶= FO S ∃ i +  ∶∶= ⊆ atoms, S i φ ∨ η ∃ ¯ for φ , η ∈ S i +  xφ φ ♭ ∧ η ∀ ¯ for φ ∈ S i , η ∈ S i +  xφ ♭ S ∀ i +  ∶∶= ⊆ atoms, S i φ ♭ ∨ η ∃ ¯ xφ ♭ for φ ∈ S i , η ∈ S i +  φ ∧ η ∀ ¯ for φ , η ∈ S i +  xφ S i +  ∶∶= S ∃ i +  ∪ S ∀ i +  Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 6 / 10

Te fragment. Take  — with negation N ∶∶= literals, ⊆ atoms φ ∨ η ∃ ¯ xφ for φ , η ∈ N φ ♭ ∧ η ∀ ¯ xφ ♭ ¬ φ ♭ φ ♭ ( ¯ x ) ∶= ∃ ¯ z ( ¯ x ⊆ ¯ z ∧ φ ( ¯ z )) with Semantics A ⊧ X φ ♭ ⊧ Y φ ♭ ∶ ⇐ ⇒ A / for all Y ∩ X ≠ ∅ N k : fragment with fewer than k negations. Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 7 / 10

N vs Solitaire-GFP Proposition Winning solitaire games of level k can be described in N k . On arbitrary structures: N k ≡ Solitaire-GFP with alternation level k . Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 8 / 10

Expressiveness Over arbitrary structures: S collapses to S ∃  = N  F ≡ S ≡ N  N i ⊊ N i +  for all levels i Over finite structures: N collapses to N  F ≡ S ≡ N Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 9 / 10

Complexity Proposition 1 Model checking F , S , or N is NL-complete (data complexity). 2 Every NL-property of ordered structures is expressible in F . Dietmar Berwanger (CNRS) An NL Fragment for Inclusion Logic June 2015 10 / 10